Then [1]} shows that for any \(\alpha >0\) ,
there exists a nonincreasing function \(\phi \)  on \([r_0,\infty )\)  such that
\(\lim _{r\rightarrow \infty }L(r)/(r^{\alpha }\phi (r))=1\) .
Therefore,
there exists \(c_{\alpha }>0\)  such that for any \(\kappa >1\) , \(s\ge r_0\)  and \(t\ge r_0\)  with
\(s\le t\le \kappa s\) ,
\(L(s)=s^{\alpha }\frac{L(s)}{s^{\alpha }}\ge c_{\alpha }\left(\frac{t}{\kappa }\right)^{\alpha }\frac{L(t)}{t^{\alpha }}=\frac{c_{\alpha }}{\kappa ^{\alpha }}L(t).\) 
